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arxiv: 2605.20617 · v1 · pith:MJGCEBSInew · submitted 2026-05-20 · 🧮 math.DS

Complete realization of multifractal entropy spectra

Xiaobo Hou , Xueting Tian This is my paper

Pith reviewed 2026-05-21 02:55 UTC · model grok-4.3

classification 🧮 math.DS
keywords multifractal entropy spectracontinuous potentialsdynamical systemsrealization theorempressure functionstopological entropyLegendre transformmixing subshifts of finite type
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The pith

Dynamical systems with topological entropy H realize any continuous concave multifractal entropy spectrum via a continuous potential.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a complete realization result: given any H greater than zero and any continuous concave function on a compact interval that reaches its maximum value H at exactly one point, every system in a broad class with topological entropy H admits a continuous potential whose multifractal entropy spectrum equals that function exactly. This shows that the possible shapes of such spectra are fully flexible once basic concavity and uniqueness conditions are met. The same spectrum can be achieved by infinitely many distinct potentials that are not related by cohomology. The authors further apply the result via Legendre duality to obtain flexibility for pressure functions defined on all real numbers, settling a prior question.

Core claim

For every H>0 and every continuous concave function on a compact interval with maximum value H attained at a unique point, each system in this class with topological entropy H admits a continuous potential whose multifractal entropy spectrum is exactly that function. The same spectrum can moreover be realized by arbitrarily many pairwise non-cohomologous potentials.

What carries the argument

The multifractal entropy spectrum of a continuous potential, which records the topological entropy of the level sets where the Birkhoff averages of the potential take each value alpha.

If this is right

  • The same spectrum can be realized by arbitrarily many pairwise non-cohomologous potentials.
  • Entropy spectra are lower semicontinuous in the Hausdorff graph metric on non-trivial mixing subshifts of finite type.
  • Upper semicontinuity of entropy spectra fails densely for non-trivial mixing subshifts of finite type.
  • Pressure functions defined on the whole real line are flexible via Legendre transform duality with the realized spectra.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that thermodynamic formalism permits complete control over the distribution of local entropies, which may extend to other multifractal quantities such as dimension spectra.
  • One could check whether analogous realization holds for concrete systems outside the stated class, such as certain interval maps with specification.
  • The flexibility theorem for pressures raises the question of whether similar statements hold when the potential is restricted to a fixed regularity class.

Load-bearing premise

The dynamical systems belong to a sufficiently rich class that allows explicit construction of continuous potentials realizing any prescribed concave spectrum.

What would settle it

A concrete mixing subshift of finite type with topological entropy H together with a continuous concave function peaking at H for which no continuous potential produces exactly that multifractal entropy spectrum.

read the original abstract

We prove a complete realization theorem for multifractal entropy spectra of continuous potentials on a broad class of dynamical systems. More precisely, for every $H>0$ and every continuous concave function on a compact interval with maximum value $H$ attained at a unique point, each system in this class with topological entropy $H$ admits a continuous potential whose multifractal entropy spectrum is exactly that function. The same spectrum can moreover be realized by arbitrarily many pairwise non-cohomologous potentials. We also study the potential dependence of entropy spectra in the Hausdorff graph metric, proving general lower semicontinuity and dense failure of upper semicontinuity for non-trivial mixing subshifts of finite type. Finally, as an application of the realization theorem, we use the Legendre transform duality between multifractal entropy spectra and pressure functions to derive a pressure flexibility theorem for pressure functions defined on the whole real line, thereby giving an affirmative answer to a question of Kucherenko and Quas~\cite{KQ2022}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves a complete realization theorem for multifractal entropy spectra of continuous potentials on a broad class of dynamical systems. For every H>0 and every continuous concave function on a compact interval with maximum value H attained at a unique point, each system in this class with topological entropy H admits a continuous potential whose multifractal entropy spectrum is exactly that function. The same spectrum can be realized by arbitrarily many pairwise non-cohomologous potentials. The paper also proves general lower semicontinuity and dense failure of upper semicontinuity for entropy spectra in the Hausdorff graph metric for non-trivial mixing subshifts of finite type, and derives a pressure flexibility theorem answering a question of Kucherenko and Quas.

Significance. If the central results hold, this is a significant contribution to multifractal analysis in dynamical systems. The complete realization of any prescribed continuous concave function as an entropy spectrum, with flexibility in the choice of potential, provides a strong existence result that clarifies the range of possible spectra. The application to pressure functions on the whole real line resolves an open question. The paper credits the use of standard entropy and pressure properties without ad-hoc parameters, and the results are falsifiable through explicit constructions in specific systems like subshifts.

minor comments (3)
  1. [Abstract] The abstract refers to 'a broad class of dynamical systems' without specifying the properties; a brief indication of the key assumptions (e.g., mixing subshifts of finite type with specification) would improve clarity.
  2. [Introduction or main theorem statement] The Hausdorff graph metric is mentioned in the context of semicontinuity results; it should be defined or a reference provided upon first use.
  3. [Bibliography] The citation to Kucherenko and Quas (KQ2022) should include the full bibliographic details for completeness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of the main results and the evaluation of its significance. The recommendation for minor revision is noted. However, the report lists no specific major comments under the MAJOR COMMENTS section.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents an existence theorem for realizing prescribed continuous concave multifractal entropy spectra via continuous potentials on a broad class of dynamical systems (mixing subshifts of finite type with specification). The argument proceeds by explicit construction of potentials whose associated pressure functions have the target Legendre transform, then verifies that the level-set entropies recover the prescribed function. This relies on standard thermodynamic formalism (pressure-entropy duality) and the flexibility of the system class, without any fitted parameters, self-definitional loops, or load-bearing self-citations. The pressure flexibility corollary follows directly from the same duality and answers an external question from Kucherenko-Quas without reducing to prior work by the present authors. The semicontinuity results are independent corollaries. No step equates a prediction to its input by construction or imports uniqueness via self-citation chains.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard background results in topological dynamics and thermodynamic formalism with no free parameters, no new postulated entities, and only domain-standard axioms.

axioms (2)
  • standard math Topological entropy and pressure are well-defined for the continuous potentials on the systems considered.
    Invoked throughout the statement of the realization theorem and the pressure application.
  • domain assumption The Legendre transform duality between multifractal entropy spectra and pressure functions holds for the systems and potentials in question.
    Explicitly used to derive the pressure flexibility theorem from the realization result.

pith-pipeline@v0.9.0 · 5701 in / 1331 out tokens · 49837 ms · 2026-05-21T02:55:18.498258+00:00 · methodology

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